Question

How does the wavelength of a helium ion compare to that of an electron accelerated through the same potential difference? (a) The helium ion has a longer wavelength, because it has greater mass. (b) The helium ion has a shorter wavelength, because it has greater mass. (c) The wavelengths are the same, because the kinetic energy is the same. (d) The wavelengths are the same, because the electric charge is the same.

Short answer

(b) The helium ion has a shorter wavelength, because it has greater mass.

Step-by-step solution

Identify the Formula for Wavelength

The de Broglie wavelength formula is $\lambda =\frac{h}{p}$, where $h$ is Planck's constant and $p$ is the momentum of the particle.

Determine Momentum from Kinetic Energy

Both particles (the helium ion and the electron) are accelerated through the same potential difference, so their kinetic energy $KE$ is equal to $qV$, where $q$ is the charge and $V$ is the potential difference. Momentum is given by $p=\sqrt{2mKE}=\sqrt{2mqV}$.

Compare Masses to Obtain Wavelength

Since both particles are accelerated by the same potential difference, use $m_e$ and $m_{\text{He}}$ for the electron and helium ion masses respectively. The electron has smaller mass compared to the helium ion, meaning its momentum is lower, leading to a longer wavelength as $\lambda =\frac{h}{\sqrt{2mqV}}$.

Evaluate the Impact of Mass on Wavelength

As mass increases, for the same kinetic energy, momentum increases, leading to a shorter wavelength. Therefore, the helium ion with the greater mass will have a shorter wavelength than the electron.

Key concepts

Momentum

Understanding momentum is crucial when exploring the behavior of particles like electrons and helium ions. In classical mechanics, momentum is defined as the product of an object's mass and velocity. For our context with particles that exhibit wave-like properties, de Broglie's equation comes into play, linking momentum to wavelength. The equation for de Broglie wavelength is given by $\lambda =\frac{h}{p}$, where $h$ is Planck's constant and $p$ is the momentum. Momentum in this scenario is not just a simple product of mass and velocity, but it's influenced by the kinetic energy when particles are accelerated by a potential difference like in the exercise above.- Remember that for any particle, regardless of size, momentum increases with kinetic energy. - A higher momentum means a shorter wavelength, which partially explains why different particles under the same conditions can have different wavelengths.

Kinetic Energy

Kinetic energy plays a pivotal role in determining a particle's behavior under motion, especially when linked with de Broglie's wavelength. For particles accelerated through a potential difference, their kinetic energy, $KE$, is determined by $KE=qV$. Here, $q$ represents the charge of the particle, and $V$ is the applied potential difference.When dealing with particles like helium ions and electrons, it's crucial to remember that while the potential difference may be the same, their masses and potential resulting speeds differ. - This results in different momenta and therefore different wavelengths, even though their kinetic energies are derived similarly. - As kinetic energy is directly used to calculate momentum $p=\sqrt{2mKE}$, the difference in mass between particles directly affects their respective wavelengths.

Mass Effect on Wavelength

The mass of a particle has a significant effect on its de Broglie wavelength. Essentially, greater mass results in a greater momentum when kinetic energy is constant. Looking at our equation, $\lambda =\frac{h}{\sqrt{2mqV}}$, we can see the impact of mass on a particle's wavelength.When a helium ion and an electron are both subjected to the same potential difference:- The helium ion, being more massive than the electron, results in a larger value under the square root (in the denominator). - This leads to a smaller calculated wavelength for the helium ion compared to the electron.It's important to note that as the mass increases, the momentum of the particle increases, inevitably reducing the wavelength. This explains why, under identical conditions of acceleration, the electron, with its much smaller mass, has a longer wavelength compared to the heavier helium ion.